More than 10,000 Free Syrian Army fighters are part of the Turkish military’s offensive, which began on Jan. 20 to oust the Kurdish-led Syrian Democratic Forces from the town of Afrin, which borders Turkey in northwestern Syria.

The Turkish government, military and official media did not initially comment on the episode.

The body was identified by Kurdish forces as that of Barin Kobani, a military pseudonym, though a spokesman declined to confirm her real name. She and three other members of the Kurdish women’s militia, the Women’s Protection Units, were killed last week defending Afrin, the militia said.

“In such a bestial scene, it is proven to the world the identity of the real invaders and the terrorists in Afrin,” the militia said in a statement. “The Turkish state is the world’s sponsor of terrorism.”

The comment on the videos that most incensed Kurds in Syria, judging from posts on Kurdish Facebook and Twitter accounts, was “She’s beautiful, man.” Her body at that point was partly naked, and her exposed breasts appeared to have been mutilated. One of the men then stood with his booted foot on her left breast.

Photo

A Turkish military convoy in Aleppo Province, Syria, on Monday. The forces are leading an assault on a Kurdish enclave in the province with the help of a Syrian militia, the Free Syrian Army.Credit
Aaref Watad/Agence France-Presse — Getty Images

“This is the spoils of war from the female pigs of the P.K.K.,” one of the men in the videos says, referring to the Kurdistan Workers’ Party, a Kurdish group from Turkey designated by the United States and Europe as terrorists.

The Syrian Kurdish militia, known as the People’s Protection Units or Y.P.G., is not considered terrorist by Western countries and says it has no affiliation with the P.K.K.

The Turkish newspaper Vatan described Barin Kobani as a would-be suicide bomber who detonated her explosives when spotted by Free Syrian Army members. The newspaper dismissed claims that the members of the Free Syrian Army tortured Ms. Kobani as “propaganda.”

The spokesman for the main Kurdish militia in the region, Birosk Hasakah, disputed that Ms. Kobani was a suicide attacker. He said that she and three other Kurdish fighters had been surrounded by Free Syrian Army forces in Qarnah village on Jan. 30 but refused to surrender.

“Barin and the others, they fought to the end,” he said. “She didn’t make any suicide.”

In the first video, 60 seconds long, Ms. Kobani’s body is laid out on her back in the dirt and appears intact, in bluejeans and wearing what appear to be ammunition vests, also apparently intact.

Nine armed men in assorted uniforms and civilian clothing pick at her corpse, searching her body and finding a grenade and an accessory pouch for her BKC light machine gun. Several times the men pose with their boots on her body. One man posed twice for a selfie with her body behind him.

In the second video clip, 13 seconds long, the victim is partly unclothed and mutilated, again laid out on her back but apparently in a different location; the color and type of dirt has changed. After the fighter put a booted foot onto her breast, a commander apparently tried to intervene. “No no no, guys,” he said.

The National Council of Syrian Revolution and Opposition Forces, a pro-Turkish group, also called for an investigation into the episode. “Barin’s body was mutilated by monsters,” Burhan Gallioun, a critic of the Y.P.G. and the former president of the Syrian National Council, wrote on Facebook. “Whoever did this to Barin does not belong to the human race.”

Mr. Abdul Rahman of the Syrian Observatory called the behavior of the men in the video horrific even by Islamic State standards. “This is more brutal, more ugly even than ISIS; they didn’t mutilate bodies like this, put their feet on someone’s chest,” he said. “They beheaded people, but they didn’t do this sort of thing.”

Mutilating dead bodies is a war crime under the international laws of armed conflict.

Some Turkish commentators praised the events depicted in the video, and this post was typical of many on social media: “If you were men, you would not give weapons to women and put them on the front lines.”

Many Kurdish women saw the conduct shown in the video as expressing hatred for the role women play in the Kurdish militias, where women and men have separate units but fight together.

“They want to break the women, they don’t want them to be free,” said Berivan Hesen, the co-president of the Kurdish civil government in Kobani, Syria. “But we’ve been following public opinion, and every woman in the world was hurt by what happened to Barin Kobani.”